Starting from the basic Hamiltonian formulation of the water wave problem with constant vorticity as proposed by [1]}, [2]},
we derive a Hamiltonian version of the Dysthe equation (a higher-order NLS equation)
for the nonlinear modulation of two-dimensional gravity waves on deep water, in the presence of a background uniform shear flow.
The resulting model exhibits a well-defined symplectic structure and conserves an energy (i.e. the reduced Hamiltonian) over time.
Our methodology, introduced recently for two- and three-dimensional irrotational gravity waves [3]}, [4]}, [5]},
consists in performing a sequence of canonical transformations that involve a reduction to normal form
(devoid of non-resonant triads) and use of a modulational Ansatz together with a scale separation lemma.
A novelty of our approach is a direct reconstruction of the surface variables from the wave envelope
through inversion of the third-order normal form transformation.
This reconstruction requires solving an auxiliary Hamiltonian system of PDEs, for which we provide an explicit derivation.
Such a procedure differs from the classical one where physical quantities like the surface elevation
are reconstructed perturbatively in terms of a Stokes expansion.
As a consequence, both steps (solving for the wave envelope and recovering the surface elevation)
consistently fit within a Hamiltonian framework.
